One of the biggest problems facing college-bound students today is the increasingly bad math education they are getting from their schooling before college. (I can't find a link backing that up, but by now it's virtually common knowlege.)

I wanted to write this, so that perhaps people in this boat would find it, and get a new perspective on the problem. I'm not a mathematician... and to the people stuck in this boat, that ought to sound like music to their ears. I'm a computer scientist. Basically, that's when you really, really like math, but really, really like applying it. That keeps you down to earth... most of us at least. So, I can give you a non-mathematician's view on math, while still knowing enough about math to know I'm not going to screw you up. (Believe me, that's a delicate balance.)

First, let's review the problem(s). I'm going to make sweeping generalizations about the state of math education. Hopefully, at least once in your life, you had a teacher for which the following isn't true. Many of you haven't.

Your teachers don't know mathematics. For the most part, they know just enough to be dangerous.

Let's follow a teacher through their requirements to graduate with a degree in teaching with a specialty in mathematics:

• Calculus & Linear Algebra: Certainly the course descriptions may look intimidating, but the fact of the matter is the majority of this is covered in many advanced high-school classes. Remember that most of your teachers are just like most of you. When they were in college, they probably bumbled their way through these courses. That means that, like you, our hapless teacher-to-be may have bumbled their way through and "managed" a passing grade. (Sound familiar?)
   
• Statistics: I took the advanced version of college statistics. It was still mostly covered in high school. Apparently the normal course is almost exactly what a high school stats course is supposed to teach. In other words, if our teacher-to-be learned it when they took it in high school, this course will teach them virtually nothing. The homework will be harder... hopefully... and that's about it.
   
• Axiomatic Geometry: This is our teacher-to-be's only chance of learning true math. Since it's only one course, the average teacher will bumble through this, just like you've bumbled through math courses in the past, and successfully fail to learn anything truly useful about math. After all, after 15 years of math education just like the one you got, a three or four credit college course just isn't enough to clean up the mess.
   

Let me summarize for you. Your teachers, by and large, are just like you: They know just enough math to scrape by the required courses. And let's face it, that was years ago. They may know even less now then they did then, because every book they've had since then was the Teacher's Edition, and we all know what that means: Answers! Their real math skills have probably atrophied.

(I want to remind you I'm making sweeping generalizations. Hopefully, you've experienced a teacher who knows math, despite all the obstacles. Unless you were lucky, not all of your teachers are exceptions.)

Your teachers had two basic kinds of professors in college:

• Professors for their teaching courses: Many of these profs barely have a grip on reality. (Ever looked at some of the crazy "education" ideas floating around? They all came from one of these dudes/dudettes.) They are teaching your teachers how to 'handle' children, what they need to know to be certified, and making them write paper after paper on educational theories... which, since your teachers are just like you and me, tend to be ignored right after the class is over. (Overall, that's good. Maybe you've had the misfortune of having a teacher who actually believes one of these crazy theories. While the experience was probably memorable, I bet you didn't learn much. It happened to me twice, and I know I didn't learn much.)
  
  These people extend the old adage: "Those who can, do. Those who can't, teach. Those who can't even manage that teach how to teach."
  
  (Of course, there are exceptions. There are some incredibly good teaching professors, just as there are incredibly good teachers. But again, they're the exceptions. Since the teaching profession as a whole tends to reward whacked-out ideas over good ones any day, the truly good teaching professors will rarely be the most famous and recognized faculty member.)
   
• Professors for their math courses: You could hope that they might learn something here. Sorry. The classes your teachers were required to take are for the most part taught by the math professors who had the least pull in the department, and thus were not able to get out of teaching the undergraduate courses. See, the professors prefer to teach courses to the grad students, which are more fun for the professors, but have course descriptions like "Linear multi-step methods and single step nonlinear methods, such as Runge-Kutta methods, for initial value problems and an introduction to questions of consistency, stability and convergence. Methods for stiff equations. Shooting methods, finite difference methods, and finite element methods for boundary value problems."
  
  ... I'll just let your eyes unglaze here ...
  
  Well, the math professors think that's fun. Anyhow, since everyone wants to teach the grad courses, that means the profs with a low quality of mathematical understanding teach the undergrad courses. Which means that these prof's classes tend to not be a good place to learn what math is. Sometimes you'll get a good teacher, but without understanding in the first place, what will they teach? Probably their own misconceptions. You should understand this by the end of the essay, because it's exactly what you've been up against: People teaching their own misconceptions.
  
  (Again, exceptions exist. On the other hand, it can also be even worse then I'm portraying: If you go to a big enough university, you might get a class taught by a TA, which is "a person who doesn't even have a master's degree yet". Oi!)
   

The upshot is that we've moved way past "the blind leading the blind"; you're actually in a situation where the blind are leading the blind, and other blind people are loudly telling them everything's fine, or not saying anything at all (in the case of college math professors). You at least know you don't know math. I wish the rest of the people involved were so honest.

Whoa... I really am claiming that your teachers don't know math, aren't I? I bet you thought I was going to wiggle out of it or defend them somehow. Nope. I'd say a lot of teachers, probably the majority, don't know math.

Then... what exactly are they teaching?

They are basically teaching arithmetic, with some stuff tacked on the side as an afterthought.

Unfortunately, the reality is the other way around.

You have been taught, if not directly then indirectly, that math is about numbers. Adding numbers. Subtracting numbers. Dividing number. Multiplying numbers. Raising number to the powers of other numbers, but only if the situation is truly desperate.

After 6-10 years of "math" class consisting entirely of these numbers, you are suddenly thrown into the world of algebra. Odds are what happened here is you really got hung up on the arithmetic idea of "equality", never absorbed properly what "equality" means in algebra, and it all went downhill from there. (We will try to fix this a little later.) If you're in college and you still don't really understand why "substitution" can be used to solve a system of equations, then this is what happened to you.

You got by by memorizing formulas, not understanding why they sometimes didn't work, applying recipes to story problems you didn't understand (either the problems or the formulas), partial credit, and a heathly dollop of luck. You never understand anything before you're doing the next thing. Everything seems completely disconnected from everything else, not to mention the real world, and you don't know why you're bothering with all this math crap.

If this doesn't seem to apply to you, then you can stop reading. But if this does apply, then I think can help you. First, let me explain to you what math really is.

Mathematics is a building process. We start from a foundation, and we see how far we can go.

If you learn enough math, you'll understand that the seperate branches of math are really all different aspects of the same thing. But for now, it will be acceptable to imagine that there are several different foundations, with several houses on top of them. One of them is called "Euclidean Geometry", and you almost certainly got at least some of that before college. Hopefully you've had at least a quick tour of the ground floor of "Set Theory", though you likely didn't notice. Another, rather small one, is called "Arithmetic". Down the road there's some scary ones like "Topology", "Discrete Mathematics", and "Algebraic Ring Theory". Forget about those for now.

Right now, probably the only thing you feel even remotely confident about in the previous list is "Arithmetic", and that because you have a pocket calculator that is better then you will even be at arithmetic. Well, I have good news and bad news: The good news is that it's perfectly OK to use the calculator and have confidence in it, because as you learn math, you will learn how to treat the calculator. (It can be tricked, and it does on occasion bend the truth a bit.) The bad news is that the reason that's OK is that Arithmetic isn't really math.

Unlike the real world, where big buildings need big foundations, mathematicians try to build the smallest foundations necessary. This is more like machines in the real world: Smaller things have fewer parts that can go wrong.

It happens that numbers provide an amazingly flexible foundation, with many applications in the real world, and with many real-world examples. However, numbers are not necessary to do math. Topology deals with shapes. Graph theory deals with dots and lines. Geometry deals with all kinds of things like circles and squares and triangles and lines and stuff. Discrete math works with all kind of constructs, including something that provides the basis for all computers. Certainly we use numbers when convenient in those math systems, but they are not the main focus per se.

Starting from these tiny foundations, mathmaticians over the millenia have constructed beautiful structures, brick by brick, floor by floor. You may feel overwhelmed staring up at Calculus as if it had fifty floors, but it wasn't always that big. In fact, what you'll have to learn will really only be the first five or six floors. That may not seem like much, but those floors hold up the entire rest of the structure. Once you understand the floor plan for the first few floors, the rest are much easier. Math is often very hierarchial like this.

This hierarchiac natures leads directly to the first, and often the last, mistake teachers and students make when learning math. I can not emphasize this enough: If you don't have a firm foundation, you cannot hope to build (learn) the higher levels. You can't skip the ground floors, anymore then you can walk into the the Sears Tower on the 20th floor. This is where you get into trouble with the modern teaching system. Your teachers don't understand math well enough to understand just how deeply this is true. But they do know what the state or federal government requires them to get through, and by golly, they will teach everything in the course, come hell or high water!

Which tends to come, often around chapter four.

You'd almost always be better off slowing down in the early parts of any given course, nail the fundamentals down until you can answer tricky problems about the fundamentals in your sleep, and only then move on to the rest of the course. After that, the rest of the course can move relatively quickly. Since you can't control the speed of the course, that means you're going to need to put a lot of effort into the beginning of the course, so you "virtually" slow it down.

Excuse me, I need to slap you.

You can learn math. In fact, if you apply yourself, math can become one of your easiest classes. If you understand the foundations, the rest will actually flow relatively easily. Honest!

But.

*slap*

I can't help you if you don't apply yourself. It is your responsibility to learn math. It always has been. Unfortunately, some people were basically lying to you and told you that you could learn math by sitting in class and having it fed to you. And they fed it to you poorly to boot.

*slap*

Nobody can teach you math. But more to the point, nobody can stop you from learning it. You are in control! That's both a sobering responsibility and a remarkable release.

Before I get into the specifics of learning about particular math branches, I want to lay out explicitly what it means that you need to learn the foundation. If you're taking college math classes and you want to do well, it is imperative that you take extra time at the beginning three or four weeks of the course to nail down the foundations of whatever you are learning. I'll talk specifics later, but resign yourself to this right now.

If you do not, you will repeat the experience you've had in every math class you've ever taken. (How's that for a threat?) Within a month or two, while you could in theory catch up, you will be right back on the same-old, same-old track.

Another major problem you may experience are ingrained bad habits. Unfortunately, I can't guess the exact bad habits you may have. All I can say is you must try to have good habits. Be careful. Check every step. Walk before you run. You will probably need to do extra problems in the first few chapters. You will eventually run, but you must earn it.

Finally, you have one last problem, but it's the worst of all: You've probably lost confidence in your ability to do math. I assure you, you can do it. Some people are better then others, and it will be easier for some then others, but I promise you that you can pass these courses. Indeed, don't think you can use the excuse that it's just "not easy" for you. While that's true for some people, it's more likely that you aren't working smart enough. I'm sure you're working plenty hard enough, but nobody ever showed you how to work smart. Let me try to help you with that with some examples.

I'm going to give some examples of the foundations I'm talking about, with real examples.

By "Basic Basic Arithmetic" I mean simple integers being added, subtracted, multiplied, and divided.

Let me be blunt. If you have to reach for a calculator to figure out what 10 divided by 2 is, you are in seriously deep shit. Let no-one candy coat this for you. You are way in the hole and effort will be required on your part to climb out.

The good news is that the rest of mathematics, despite what you may have been taught, isn't really about memorization. The bad news is that basic arithmetic is. There is no way around that. All clever attempts to avoid this unpleasent fact have backfired horribly.

You must know the multiplication tables out to ten times ten. The more you know, the better. (Fifteen times fifteen is doable.) You must know what 6 times 8 is without figuring it. You must know how to add four three digit numbers. There are no shortcuts around these facts. If you still need to work on these things, then you are going to have to do it yourself.

Why is this so importent? Three major reasons:

1. Your calculator isn't always correct... because you frequently make typos. If you're going to do a calculation, you really need at least a vague idea of the right answer, so if you type in one too many digits, you can catch it when the answer is really wrong. Without this, you will bleed points on every quiz and every exam you use a calculator on.
    
2. Independence & confidence. If you can do arithmetic in your head without resorting to electronic aid, you will increase your confidence. If you can't, and especially if you are losing points on the exam every time, you will lose confidence. That's bad, and the loss of confidence will build over time.
    
3. Speed. This is the biggie. Calculators slow you down when you use them to do arithmetic you should already know. Calculators slow you down by another factor of three or four when you don't know enough arithmetic to know when you can trust your calculator, or when you made a typo (see #1). You will waste hours or days worth of time on this per semester, if things are bad enough.
    

The only thing you can do is drill & memorize. Numerous computer programs exist to help with this. They don't make it fun, but they may make it palatable. Best of all, they prevent you from cheating yourself. I don't have specific recommendations, but I'd look for one that gives you specific feedback about speed, accuracy, and how you're comparing to yourself over time. You can watch yourself improve, and you won't be able to lie to yourself. (Haven't you heard enough lies?)

Algebra has been around for hundreds of years. The first time you see things with letters instead of numbers, they call it algebra. I'm going to assume you've seen it by now, so I don't need to explain to you what variables are from scratch. I also assume that you know, or at least can look up, the order of operations. Instead, I can just make some corrections :-)

Actually, you've been doing algebra instinctively almost since you could add 2 and 2. Any time you've made change or figured out how to fairly divide a pie, you've been doing algebra.

Algebra as taught in high school and early college basically consists of exploring the surprising consequences of a symbol you probably think you understand: =. Equality. Equality is one of the basic foundations of mathematics, and it's importent that you understand what it means.

This, and a handful of other basic rules, form the foundation of what we call algebra. I'll get to the other rules in a moment. Let's talk about equality, because without that, you're toast for all the rest of mathematics.

Most people think, and I imagine most people teach, that the equality symbol means that the two things on the left and the right of the symbol are the same number. Thus, the following:

Would mean that x "is" 2. There's nothing incorrect about that... we'll be hearing that a lot as we tour the fundamentals... but there's a much better way of thinking of it.

What the equality symbol really means is that the two things are freely substitutable for one another. Think about the difference between that and your current understanding. For the above expression, that means that anywhere you see an x, you can substitute a "2". Equivalently, which most people miss, anywhere you see a 2 you can replace it with an x.

This is one of the fundamental powers of mathematics: To find two things that are "so" the same, that they can be freely substituted for one another.

Slightly less trivial:

Now, anywhere you decide you want the area of a circle, you can replace it with "pi*radius²".

Let's look at the first example again.

Now, let's get silly for a moment. Like we said, anywhere we see an x, we can replace it with a 2. So, let's go ahead and do it.

Well, I hope you consider THAT obvious! ;-)

I want to use this to explain algebraic manipulations. By now, you know that "solving" expressions involves subtracting things from both sides, and adding things, and all kinds of crazy things that you may find hard to remember. (Fortunately, you don't really have to.)

For the moment, we've made it incredibly obvious that both sides of the equation are precisely the same thing. Now, the fundamental principle of algebraic manipulation is very simple: If we do the same thing to two things that are exactly the same, we get will get two other things that are also exactly the same.

Stop. Do not pass this sentence until you understand that. Take two copies of the same thing, do the exact same thing to both of them, and you get two identical results. Sounds stupid, but it's one of the basics of reason itself. Let me call that statement zero:

0. (English Version) If you do the same thing to both sides of an equality, the two results are still equal.

0. (More Mathy Version) If x = y, and f() is some function, f(x) = f(y).

(You might want to take a moment and try to understand how the "mathy" version relates to the "English" version. That skill will serve you well when it comes time to read your math books.)

(I'd like to point out that there ARE exceptions, but you need to understand the previous statement before you can understand the exceptions.)

Now, this may seem silly, but let's play a bit with the 2's for a moment. On the left are the equalities we have. (You may be used to hearing that called an "equation". That is of course the proper word, but it tends to obscure the fact that what we have is just an "equality". Nothing magic. It's just Latin screwing with you again.) On the right is what I did. (This is a very normal way to show mathematical manipulations.)

Now, I know that looks stupid. Of course "(2 + 1) / 3" equals "(2 + 1) / 3"! But let's put the x back...

Note that I left the second comment alone. There are still two 2's on that line... it's just that one happens to look like an x. So why update the text? It's still accurate.

At the end of our little exercise, we had this expression:

Now, you're probably SO used to computing that in your head that you want me to replace it with "1". (Touch your calculator for that and I'll be very annoyed.) But I want to take one last opportunity to bash this equality thing in to your head before moving on.

It is not necessarily immediately obvious that I can replace (2 + 1) / 3 with 1. It only seems that way because you are a smart individual who has basically been working with numbers all your life. (You may not feel that way, but it is true.) The reason you can do that replacement is that the following is true:

See, there's that = sign again. Because 1 is fully replaceable by (2 + 1) / 3, and (2 + 1) / 3 is fully replaceable by 1, I can take the (2 + 1) / 3 and replace it with 1.

Not a very profound statement, but it's a true one.

All other equation solving works on similar lines.

The thing you will do most often on Algebra homework and Algebra exams is "solve equations", which is a two word phrase guarenteed to strike fear into the hearts of the hardiest of jocks. But there's nothing to be afraid of here.

One of the great truths of mathematics is that there is no One Correct Representation of anything. Consider the number 2, which we've been abusing for a while already. You're used to 2. But 2 also equals 1 + 1.

Remember what that means? Anywhere we see two, we can replace it with 1 + 1.

For that matter,

There are an infinite number of ways to express the idea of "2". ("Inifinite" is the mathematician's way of saying that you couldn't possibly write all of them down, no matter how quickly you wrote them, in any amount of time. There are better definitions, but worry about those when you get to a class that uses them.)

The other thing you need to wrap your mind around to "get" algebra is that no one of them is more "right" then another. Remember what I said equality is? Anywhere you see one representation of 2, you can replace it with another. You may prefer "2" over "(800/200) - 16 + (7*2) - cos(0)", but that's your human preference, not a law of mathematics. Both express the same quantity.

So it is with equations. For any equation, there are an infinite number of other equations that are fully equivalent. When you are asked to "solve" for x, your teacher/prof is asking you to give him/her the version of the equation that has the x all alone on one side, and as few symbols as possible on the other, which is usually called "simplification".

To finish this discussion, I need to show you the other fundamentals of algebra.

Remember how I discussed the foundations of math? I've spent a lot of time talking about equality. Equality is so fundamental, that it is in the foundation of every branch of math, including Algebra. I had to talk about equality, because the rest of the foundations of Algebra actually depend on it; it's almost like the Earth is the foundation of the foundation for buildings. Here's the foundation (which comes from the same root word as "fundamental") of Algebra:

1. x + 0 = x: Or, in English, any number plus zero is substitutable by itself.
    
2. x * 1 = x: Or, any number times one is substitutable by itself
    
3. x - x = 0: Or, any number minus itself is substitutable by zero.
    
4. x / x = 1 if x isn't 0: Or, any number divided by itself is substitutable by 1, except 0.
    

That, plus the exact definition of +, -, *, /, and exponentiation ('to the power of'), is most of the foundation of Algebra. (There's a couple of details I left out, and theres some stuff about exponentiation, but this is most of what you care about for now.) Actually telling you the whole foundation wouldn't take much longer, but I don't want to overload you with stuff you may not care about right away.

Math term: What I've been calling the "foundation" is known in math circles, and hopefully your math textbook, as axioms.

Now for the big question: Why is this importent to me? The reason is, those few things are Algebra. The rest of what you call "Algebra" is merely exploring what those statements mean, and what they tell us. There are other branches of mathematics where those statements aren't true, or don't exist. For instance, in a branch called "Graph Theory", there's nothing that you'd call "division". "Set Theory" doesn't have anything you'd really call multiplication. There are entire other branches of math (like the study of Vectors) that are controlled by completely different ideas of multiplication. Many branches have different ideas of "numbers", or don't even necessarily involve numbers at all, like Geometry.

Each of those branches of math are just as normal as Algebra, they just choose different foundations, and explore those instead.

Let us solve a simple equation, using just the axioms of Algebra.

I hope that you can immediately see that the answer is

But to arrive at that answer, you took a lot of shortcuts, more then you probably even realize. There's nothing wrong with that per se, but you really need to understand what's really going on.

Let's look at that equation for a moment.

This is a very simple equation. What do we want to do? We want to isolate the x, so that it is alone on its side of the equality symbol. So, what can we do to accomplish that goal, using just the axioms 0 - 4? You might say "nothing", but that's not right. We can always use the axioms. It's just that they don't always get us where we want to go. That's your job as the mathematician. So let's play stupid for a moment, and look at where the axioms can get us, starting with #1.

#1 says x + 0 = x. Well, nothing looks like the left side, so we can't replace things of the left form with things of the right form. But we can go the other way. We can get:

Because we are smart humans, we can look at that and realize we aren't going to get anywhere that way. There were some early computer programs that tried to solve equations that weren't even that smart, though, so be glad you're as smart as you are, and so that you don't waste time with this.

#2 says x * 1 = x. Again, same basic waste of time:

Maybe #3? x - x = 0 Well, we don't have anything that looks like x - x or anything that looks like 0 in our original equation. But, we can take one from when we used rule #1, 2x + 0 = 2, and work with that to get something like:

2x + 8839 - 8839 = 0

8839 could of course be anything we want. Now that's an importent point. Sometimes in math we get to pick whatever we want. It's importent that we pick the right thing. Remember that for a moment, it's about to get really importent...

For completeness, let's try #4: x / x = 1 if x isn't 0. It's just like before... nothing like that in the original equation, but with the fully equivalent equation 2x * 1 = 2, we can apply rule #4 from right-to-left to get something like

2x * (421.4 / 421.4) = 2

Again, 421.4 could be anything we want, except for the special case of "0", which we won't allow until Calculus, where we develop some techniques to handle certain cases of 0 / 0.

You'll note none of the rules 1 - 4 are of much use in solving equations, even the simplest ones. We need rule #0 ("If you do the same thing to both sides of an equality, the two results are still equal.") to get anywhere. But there are a lot of "same thing"s we could do to the equation. Hopefully, it's obvious to you what to do. But how do we know what to do?

That is exactly the question that what you know of as "Algebra" seeks to answer. The only rule in Algebra that is of any real use is Rule #0, and it is that rule we must use to solve equations. The others will be used only to clean up after ourselves. BUT. . . it is importent to mathematicians that we have those rules, because unless we accept as an axiom the fact that x + 0 = x, we do NOT know that it's true. You could create your own branch of math where that was false! (In fact, there are technically an infinite number of branches of math; fortunately, we only care about a few of them.)

It may seem like we're making a mountain out of a molehill, but if you just bear with me a little longer, we'll start doing some real work.

We have our little equation: 2x = 2. We want to isolate the x. Wouldn't it be nice if we could use one of the rules to get rid of the two? We could pair up 1 & 3, or 2 & 4. Clearly from your own experience, we want the multiplication rules.

How can we apply rule 4? Well, we need to get something that looks like x / x to go from left-to-right, or 1 to go from right-to-left. We want the first one, so we want something that looks like x / x.

Only now is it time to apply rule #0 and do what you've been itching to do for the last couple of pages. We now know what we want: 2 / 2. The way we'll get it is to divide both sides by 2:

And we are on our way. Imagine for the moment that the division is written the way you are used to. (Perhaps if I get up the initiative, I'll do the division up in HTML so it appears correctly.) Let's use the laws of division to re-write our expression as so:

We know we can do that because of the properties of division, which I encourage you to review but I don't want to go over. I want to point something out: We created a new expression. While (2x) / 2 = 2 / 2 is equivalent to (2 / 2) x = 2 / 2, they are not the same expression. We know they are equivalent only because of the properties of division. If it wasn't for that, we couldn't be sure.

(What does equivalent mean? Well, it can mean different things depending on context. In this case, it means that the equality is true for exactly the same values of the variables. No more solutions, no less. This becomes importent when we get to solving quadratic equations!)

Now, by rule #4 we know that anything of the form x / x is fully replaceable by 1, as long as x isn't 0. It's not 0, so we replace (2 / 2) with 1:

And hey, there's one of those (2/2) things on the other side, too:

Now, we've still got more work to do! In all likelihood, you want to take the shortcut, and just drop the 1 off the front. And like I said before, there's nothing wrong with taking shortcuts, when you understand exactly what you're shortcutting around. We do not know that we can drop the one off the front of the x. It is not a natural law of the universe. In fact, it's not even a natural law of "mathematics"! Exactly what "1 times x" means totally depends on the axioms we are using.

For that matter, what "multiplication" means also depends on the axioms you are using. In the branch of mathematics that deals with ultra-hyper-formalizations of this stuff called "number theory", we have a definition of multiplication, and we are basically borrowing that definition and silently adding it to our axiom set. You don't really care, because you "know" what multiplication is, and it's not worth going over the formal definition, because it would involve too much other stuff. But remember that multiplication as you know it is specific to numbers-as-you-know-them. (BTW, the "mathy" term for those numbers would be "real scalars".)

To complete our simplification, we must invoke rule #2, going from left-to-right. We take the "1x" and replace it with "x", in accordance with our axiom. Only now are we done:

There are two major lessons to carry away from this discussion.

First, even the simplest of tasks, solving the equation 2x = 2, requires a surprisingly large number of steps. You might wonder what a real mathematical problem might take to solve. And the answer is, if you have to grind through every step, without skipping anything, it does take millions, billions, trillions, untold gazillions of steps to solve some problems.

Every time you play an MP3 file, the computer is performing millions of mathematical manipulation per second, just to decode the song. The song is encoded in a format designed to be "easily" decoded through some amazingly clever math work done by your computer. And those operations took hundreds of years of mathematical innovation and discovery to create; on the scale of the operations we needed to solve 2x = 2 in the above example, I doubt there's anyone alive that has a clue how much mathematics that really represents.

That's the benefit of computers. They love grinding through a billion operations just so you can listen to your favorite song again. Obviously, we humans are a bit more picky and a bit slower. Thus, we create every shortcut our amazingly powerful minds can come up with. And let me tell you, as a race, we've found some real doozies.

In learning math, there's a tension. On the one hand, you really need to understand as much as possible about what you're doing, or you will get lost, which you probably don't need me to remind you of. On the other hand, if everybody had to do every single operation from scratch, as carefully as we just solved that equation, the human race would have never discovered calculus. Forget about discovering E = mc², Einstein would still be doing his tax returns today! (Or his legal heirs, at least.)

Thus, we must strike a delicate balance. We must try to learn things carefully, and only then can we skip along the shortcut. I know I've said it over and over, but hopefully every time I say it the reasoning becomes clearer: You cannot skip the foundation building. If you skip the first problems in the course because they seem easy, or because you thought you understood it last time you took the course, check to make sure you really do understand, because without the foundation, you are lost in an unbeleivably complex maze. Solving 2x = 2 wasn't even the tip of the iceberg.

You have in all likelihood been forced to advance too quickly, before you understand what's going on. (Not surprisingly, few people restore the balance by spending more time learning things carefully.) The balance is different for everybody. All I can say for certain without knowing you is that if you had to read this essay, you probably need to spend more time carefully learning. But remember the payoff: If you learn carefully, you will later get to use the shortcuts at will. The later parts of the course are doable when you have the shortcuts at your command. So, your job is to find your own pace. Sadly, there's nobody alive who can simply tell you what that pace is... you have to find it on your own.

That's lesson one. Lesson two you may find more encouraging. By now, you have probably observed more then once that all this stuff seems very arbitrary ("Determined by chance, whim, or impulse, and not by necessity, reason, or principle"). Well, congratulate yourself, you are very, very close to being right, and it's one of those truths that can truly set you free. Math is indeed totally arbitrary, only I'd choose the meaning "Depending on will or discretion; not governed by any fixed rules".

Algebra as you know it is totally created by the selection of a few choice axioms and the exercise of logic and reason. There's no magical reason for anything in Algebra, it's all in the axiom set. So indulge your feelings that this is arbitrary. It is.

But don't confuse that with "made up" or "irrelevent". We study Algebra because out of an amazingly small foundation and a handful of definitions, we get amazing, totally unexpected results. Simply staring at the axiom set (which consists of perhaps 20 to 25 axioms in a truly complete definition of Algebra) shows no hint of the Quadratic Equation or any of the other things you've studied, but it's all there. Also, using Algebra, we can describe many things in the world amazingly well. Even real world things, like business accounting. Algebra has the consensus of hundreds of years and millions of mathematicians behind it. Believe me, it's worth studying, no matter what we may think in our darker moments.

Always remember, it's a man made construct, one of many. Humans made it, and humans can understand it.

You have weathered the tough part of algebra. It's all downhill sledding from here. Now that you hopefully have a grasp of what we're really doing in Algebra class, I can unmask the monster and show you the point. You may even find yourself approving of the whole exercise!

The point of the Algebra you are studying is to answer the question "How do we know what to do to solve an equation?" by creating, studying, collecting, and using shortcuts. Yes, the entire point of Algebra is to save you time! By cleverly using our axioms, we can create new rules on the fly, and use them to save us from having to repeat calculations, over and over again. And because we create them directly from the axioms, we can have great confidence in our new rules.

Your class periods are strategy sessions on how to approach certain types of problems. You should not find this hard to see in your next class. Your homework problems are exercises for your brain, so you can drill the current shortcut in your head. Unfortunately, you can't skip this step because you are not a computer. Like any worthy endeavor, there's a lot of grunt work involved.

The cool bit is that once you make a new rule, you can make new rules out of that. You don't need a license, you don't have to be a "mathematician", anyone can do it. This is how we build the structure of math. This is the structure that famous people throughout the ages have called beautiful, amazing, incredible, and every other superlative you can imagine. If you can just once experience the flash of recognition, that this structure we call math is amazing, then it will have all been worthwhile, for you will have touched one of the foundations of the universe. Few other classes can offer that. I am truly saddened that mathematical education so fully fails to show this to you, and instead turns math into a boring, haphazard, random death march through a proscribed cirriculum.

Unfortunately, I can't offer much help when it comes to the trig functions "sine", "cosine", and "tangent". Unfortunately, the "higher structure" that would help you understand the relationships between these three functions are second-semester calculus. This is one place where I'm afraid I will have to recommend rote memorization.